LMFDB - Embedded newform 1296.5.e.g.161.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1296,5,Mod(161,1296)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1296, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 5, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1296.161");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 1296 = 2^{4} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	1296.e (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$133.967472157$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} + 6x^{6} + 121x^{5} + 1104x^{4} - 1647x^{3} + 6529x^{2} + 85254x + 440076 $ x^8 - 3x^7 + 6x^6 + 121x^5 + 1104x^4 - 1647x^3 + 6529x^2 + 85254*x + 440076
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{20} $
Twist minimal:	no (minimal twist has level 36)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.6
Root			$4.23522 - 4.06612i$ of defining polynomial
Character	$\chi$	$=$	1296.161
Dual form			1296.5.e.g.161.3

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+12.2819i q^{5} +14.2840 q^{7} +O(q^{10})$	$q+12.2819i q^{5} +14.2840 q^{7} +104.171i q^{11} +75.2346 q^{13} +341.998i q^{17} +706.329 q^{19} +596.312i q^{23} +474.155 q^{25} -1302.39i q^{29} -1029.02 q^{31} +175.435i q^{35} +563.132 q^{37} +99.1448i q^{41} +896.514 q^{43} +430.570i q^{47} -2196.97 q^{49} -5271.47i q^{53} -1279.41 q^{55} +5639.26i q^{59} +1131.25 q^{61} +924.023i q^{65} +1352.82 q^{67} +5681.42i q^{71} +4236.54 q^{73} +1487.98i q^{77} +6135.42 q^{79} +7509.88i q^{83} -4200.38 q^{85} -8721.70i q^{89} +1074.65 q^{91} +8675.05i q^{95} +5441.31 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 26 q^{7}+O(q^{10}) $ 8 * q + 26 * q^7	$ 8 q + 26 q^{7} + 10 q^{13} - 562 q^{19} - 706 q^{25} + 374 q^{31} + 16 q^{37} - 136 q^{43} + 654 q^{49} + 1818 q^{55} + 3874 q^{61} + 308 q^{67} - 7802 q^{73} - 4390 q^{79} + 6084 q^{85} - 15830 q^{91} - 14564 q^{97}+O(q^{100}) $ 8 * q + 26 * q^7 + 10 * q^13 - 562 * q^19 - 706 * q^25 + 374 * q^31 + 16 * q^37 - 136 * q^43 + 654 * q^49 + 1818 * q^55 + 3874 * q^61 + 308 * q^67 - 7802 * q^73 - 4390 * q^79 + 6084 * q^85 - 15830 * q^91 - 14564 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1296\mathbb{Z}\right)^\times$.

$n$	$325$	$1135$	$1217$
$\chi(n)$	$1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	12.2819i	0.491275i	0.969362	+	0.245638i	$0.0789974\pi$
$5$	12.2819i	0.491275i	−0.969362	+	0.245638i	$0.921003\pi$
$6$	0	0
$7$	14.2840	0.291511	0.145756	−	0.989321i	$-0.453439\pi$
$7$	14.2840	0.291511	0.145756	+	0.989321i	$0.453439\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	104.171i	0.860916i	0.902611	+	0.430458i	$0.141648\pi$
$11$	104.171i	0.860916i	−0.902611	+	0.430458i	$0.858352\pi$
$12$	0	0
$13$	75.2346	0.445175	0.222588	−	0.974913i	$-0.428550\pi$
$13$	75.2346	0.445175	0.222588	+	0.974913i	$0.428550\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	341.998i	1.18338i	0.806164	+	0.591692i	$0.201540\pi$
$17$	341.998i	1.18338i	−0.806164	+	0.591692i	$0.798460\pi$
$18$	0	0
$19$	706.329	1.95659	0.978295	−	0.207218i	$-0.0664411\pi$
$19$	706.329	1.95659	0.978295	+	0.207218i	$0.0664411\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	596.312i	1.12724i	0.826033	+	0.563622i	$0.190593\pi$
$23$	596.312i	1.12724i	−0.826033	+	0.563622i	$0.809407\pi$
$24$	0	0
$25$	474.155	0.758648
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 1302.39i	− 1.54862i	−0.632807	−	0.774309i	$-0.718097\pi$
$29$	− 1302.39i	− 1.54862i	0.632807	−	0.774309i	$-0.281903\pi$
$30$	0	0
$31$	−1029.02	−1.07078	−0.535391	−	0.844605i	$-0.679836\pi$
$31$	−1029.02	−1.07078	−0.535391	+	0.844605i	$0.679836\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	175.435i	0.143212i
$36$	0	0
$37$	563.132	0.411346	0.205673	−	0.978621i	$-0.434062\pi$
$37$	563.132	0.411346	0.205673	+	0.978621i	$0.434062\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	99.1448i	0.0589797i	0.999565	+	0.0294898i	$0.00938826\pi$
$41$	99.1448i	0.0589797i	−0.999565	+	0.0294898i	$0.990612\pi$
$42$	0	0
$43$	896.514	0.484864	0.242432	−	0.970168i	$-0.422055\pi$
$43$	896.514	0.484864	0.242432	+	0.970168i	$0.422055\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	430.570i	0.194916i	0.995240	+	0.0974581i	$0.0310712\pi$
$47$	430.570i	0.194916i	−0.995240	+	0.0974581i	$0.968929\pi$
$48$	0	0
$49$	−2196.97	−0.915021
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 5271.47i	− 1.87664i	−0.345773	−	0.938318i	$-0.612383\pi$
$53$	− 5271.47i	− 1.87664i	0.345773	−	0.938318i	$-0.387617\pi$
$54$	0	0
$55$	−1279.41	−0.422947
$56$	0	0
$57$	0	0
$58$	0	0
$59$	5639.26i	1.62001i	0.586423	+	0.810005i	$0.300536\pi$
$59$	5639.26i	1.62001i	−0.586423	+	0.810005i	$0.699464\pi$
$60$	0	0
$61$	1131.25	0.304018	0.152009	−	0.988379i	$-0.451426\pi$
$61$	1131.25	0.304018	0.152009	+	0.988379i	$0.451426\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	924.023i	0.218704i
$66$	0	0
$67$	1352.82	0.301364	0.150682	−	0.988582i	$-0.451853\pi$
$67$	1352.82	0.301364	0.150682	+	0.988582i	$0.451853\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	5681.42i	1.12704i	0.826101	+	0.563522i	$0.190554\pi$
$71$	5681.42i	1.12704i	−0.826101	+	0.563522i	$0.809446\pi$
$72$	0	0
$73$	4236.54	0.794996	0.397498	−	0.917603i	$-0.369878\pi$
$73$	4236.54	0.794996	0.397498	+	0.917603i	$0.369878\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1487.98i	0.250967i
$78$	0	0
$79$	6135.42	0.983083	0.491542	−	0.870854i	$-0.336434\pi$
$79$	6135.42	0.983083	0.491542	+	0.870854i	$0.336434\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	7509.88i	1.09013i	0.838395	+	0.545063i	$0.183494\pi$
$83$	7509.88i	1.09013i	−0.838395	+	0.545063i	$0.816506\pi$
$84$	0	0
$85$	−4200.38	−0.581367
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 8721.70i	− 1.10109i	−0.834807	−	0.550543i	$-0.814421\pi$
$89$	− 8721.70i	− 1.10109i	0.834807	−	0.550543i	$-0.185579\pi$
$90$	0	0
$91$	1074.65	0.129774
$92$	0	0
$93$	0	0
$94$	0	0
$95$	8675.05i	0.961224i
$96$	0	0
$97$	5441.31	0.578309	0.289154	−	0.957282i	$-0.406626\pi$
$97$	5441.31	0.578309	0.289154	+	0.957282i	$0.406626\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 7482.63i	− 0.733519i	−0.930316	−	0.366759i	$-0.880467\pi$
$101$	− 7482.63i	− 0.733519i	0.930316	−	0.366759i	$-0.119533\pi$
$102$	0	0
$103$	−13576.5	−1.27972	−0.639860	−	0.768492i	$-0.721008\pi$
$103$	−13576.5	−1.27972	−0.639860	+	0.768492i	$0.721008\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 16741.7i	− 1.46229i	−0.682224	−	0.731143i	$-0.738987\pi$
$107$	− 16741.7i	− 1.46229i	0.682224	−	0.731143i	$-0.261013\pi$
$108$	0	0
$109$	−12068.7	−1.01579	−0.507897	−	0.861418i	$-0.669577\pi$
$109$	−12068.7	−1.01579	−0.507897	+	0.861418i	$0.669577\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	704.601i	0.0551806i	0.999619	+	0.0275903i	$0.00878337\pi$
$113$	704.601i	0.0551806i	−0.999619	+	0.0275903i	$0.991217\pi$
$114$	0	0
$115$	−7323.84	−0.553787
$116$	0	0
$117$	0	0
$118$	0	0
$119$	4885.11i	0.344970i
$120$	0	0
$121$	3789.45	0.258824
$122$	0	0
$123$	0	0
$124$	0	0
$125$	13499.7i	0.863981i
$126$	0	0
$127$	16050.0	0.995105	0.497552	−	0.867434i	$-0.334232\pi$
$127$	16050.0	0.995105	0.497552	+	0.867434i	$0.334232\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	15504.3i	0.903463i	0.892154	+	0.451732i	$0.149194\pi$
$131$	15504.3i	0.903463i	−0.892154	+	0.451732i	$0.850806\pi$
$132$	0	0
$133$	10089.2	0.570368
$134$	0	0
$135$	0	0
$136$	0	0
$137$	890.615i	0.0474514i	0.999719	+	0.0237257i	$0.00755283\pi$
$137$	890.615i	0.0474514i	−0.999719	+	0.0237257i	$0.992447\pi$
$138$	0	0
$139$	−9358.18	−0.484353	−0.242176	−	0.970232i	$-0.577861\pi$
$139$	−9358.18	−0.484353	−0.242176	+	0.970232i	$0.577861\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	7837.25i	0.383258i
$144$	0	0
$145$	15995.8	0.760798
$146$	0	0
$147$	0	0
$148$	0	0
$149$	40669.4i	1.83187i	0.401323	+	0.915937i	$0.368550\pi$
$149$	40669.4i	1.83187i	−0.401323	+	0.915937i	$0.631450\pi$
$150$	0	0
$151$	24974.5	1.09532	0.547661	−	0.836700i	$-0.315518\pi$
$151$	24974.5	1.09532	0.547661	+	0.836700i	$0.315518\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 12638.3i	− 0.526049i
$156$	0	0
$157$	−32918.4	−1.33549	−0.667743	−	0.744392i	$-0.732739\pi$
$157$	−32918.4	−1.33549	−0.667743	+	0.744392i	$0.732739\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	8517.75i	0.328604i
$162$	0	0
$163$	−13796.3	−0.519262	−0.259631	−	0.965708i	$-0.583601\pi$
$163$	−13796.3	−0.519262	−0.259631	+	0.965708i	$0.583601\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 11332.4i	− 0.406339i	−0.979144	−	0.203170i	$-0.934876\pi$
$167$	− 11332.4i	− 0.406339i	0.979144	−	0.203170i	$-0.0651243\pi$
$168$	0	0
$169$	−22900.8	−0.801819
$170$	0	0
$171$	0	0
$172$	0	0
$173$	39170.4i	1.30878i	0.756158	+	0.654389i	$0.227074\pi$
$173$	39170.4i	1.30878i	−0.756158	+	0.654389i	$0.772926\pi$
$174$	0	0
$175$	6772.86	0.221155
$176$	0	0
$177$	0	0
$178$	0	0
$179$	24097.3i	0.752078i	0.926604	+	0.376039i	$0.122714\pi$
$179$	24097.3i	0.752078i	−0.926604	+	0.376039i	$0.877286\pi$
$180$	0	0
$181$	10277.1	0.313699	0.156850	−	0.987623i	$-0.449866\pi$
$181$	10277.1	0.313699	0.156850	+	0.987623i	$0.449866\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	6916.33i	0.202084i
$186$	0	0
$187$	−35626.2	−1.01879
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 35030.4i	− 0.960237i	−0.877204	−	0.480119i	$-0.840594\pi$
$191$	− 35030.4i	− 0.960237i	0.877204	−	0.480119i	$-0.159406\pi$
$192$	0	0
$193$	5240.55	0.140690	0.0703448	−	0.997523i	$-0.477590\pi$
$193$	5240.55	0.140690	0.0703448	+	0.997523i	$0.477590\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	42421.5i	1.09308i	0.837431	+	0.546542i	$0.184056\pi$
$197$	42421.5i	1.09308i	−0.837431	+	0.546542i	$0.815944\pi$
$198$	0	0
$199$	−31270.0	−0.789627	−0.394814	−	0.918761i	$-0.629191\pi$
$199$	−31270.0	−0.789627	−0.394814	+	0.918761i	$0.629191\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 18603.4i	− 0.451440i
$204$	0	0
$205$	−1217.69	−0.0289753
$206$	0	0
$207$	0	0
$208$	0	0
$209$	73578.8i	1.68446i
$210$	0	0
$211$	31124.4	0.699096	0.349548	−	0.936918i	$-0.386335\pi$
$211$	31124.4	0.699096	0.349548	+	0.936918i	$0.386335\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	11010.9i	0.238202i
$216$	0	0
$217$	−14698.6	−0.312145
$218$	0	0
$219$	0	0
$220$	0	0
$221$	25730.1i	0.526813i
$222$	0	0
$223$	−16027.9	−0.322305	−0.161153	−	0.986930i	$-0.551521\pi$
$223$	−16027.9	−0.322305	−0.161153	+	0.986930i	$0.551521\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	67528.9i	1.31050i	0.755411	+	0.655251i	$0.227437\pi$
$227$	67528.9i	1.31050i	−0.755411	+	0.655251i	$0.772563\pi$
$228$	0	0
$229$	78065.8	1.48864	0.744321	−	0.667823i	$-0.232773\pi$
$229$	78065.8	1.48864	0.744321	+	0.667823i	$0.232773\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	61204.9i	1.12739i	0.825983	+	0.563696i	$0.190621\pi$
$233$	61204.9i	1.12739i	−0.825983	+	0.563696i	$0.809379\pi$
$234$	0	0
$235$	−5288.21	−0.0957576
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 43154.9i	− 0.755500i	−0.925908	−	0.377750i	$-0.876698\pi$
$239$	− 43154.9i	− 0.755500i	0.925908	−	0.377750i	$-0.123302\pi$
$240$	0	0
$241$	−56207.6	−0.967745	−0.483873	−	0.875138i	$-0.660770\pi$
$241$	−56207.6	−0.967745	−0.483873	+	0.875138i	$0.660770\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 26982.9i	− 0.449527i
$246$	0	0
$247$	53140.4	0.871025
$248$	0	0
$249$	0	0
$250$	0	0
$251$	15739.4i	0.249828i	0.992168	+	0.124914i	$0.0398655\pi$
$251$	15739.4i	0.249828i	−0.992168	+	0.124914i	$0.960135\pi$
$252$	0	0
$253$	−62118.3	−0.970462
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 58760.1i	− 0.889643i	−0.895619	−	0.444822i	$-0.853267\pi$
$257$	− 58760.1i	− 0.889643i	0.895619	−	0.444822i	$-0.146733\pi$
$258$	0	0
$259$	8043.81	0.119912
$260$	0	0
$261$	0	0
$262$	0	0
$263$	94338.8i	1.36389i	0.731404	+	0.681944i	$0.238865\pi$
$263$	94338.8i	1.36389i	−0.731404	+	0.681944i	$0.761135\pi$
$264$	0	0
$265$	64743.6	0.921945
$266$	0	0
$267$	0	0
$268$	0	0
$269$	79782.6i	1.10256i	0.834319	+	0.551282i	$0.185861\pi$
$269$	79782.6i	1.10256i	−0.834319	+	0.551282i	$0.814139\pi$
$270$	0	0
$271$	76677.1	1.04406	0.522032	−	0.852926i	$-0.325174\pi$
$271$	76677.1	1.04406	0.522032	+	0.852926i	$0.325174\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	49393.1i	0.653132i
$276$	0	0
$277$	113156.	1.47475	0.737374	−	0.675485i	$-0.236066\pi$
$277$	113156.	1.47475	0.737374	+	0.675485i	$0.236066\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	96549.7i	1.22275i	0.791341	+	0.611376i	$0.209384\pi$
$281$	96549.7i	1.22275i	−0.791341	+	0.611376i	$0.790616\pi$
$282$	0	0
$283$	−30537.1	−0.381289	−0.190645	−	0.981659i	$-0.561058\pi$
$283$	−30537.1	−0.381289	−0.190645	+	0.981659i	$0.561058\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1416.19i	0.0171932i
$288$	0	0
$289$	−33441.6	−0.400397
$290$	0	0
$291$	0	0
$292$	0	0
$293$	4054.67i	0.0472303i	0.999721	+	0.0236151i	$0.00751763\pi$
$293$	4054.67i	0.0472303i	−0.999721	+	0.0236151i	$0.992482\pi$
$294$	0	0
$295$	−69260.7	−0.795872
$296$	0	0
$297$	0	0
$298$	0	0
$299$	44863.3i	0.501821i
$300$	0	0
$301$	12805.8	0.141343
$302$	0	0
$303$	0	0
$304$	0	0
$305$	13893.9i	0.149357i
$306$	0	0
$307$	−44297.5	−0.470005	−0.235002	−	0.971995i	$-0.575510\pi$
$307$	−44297.5	−0.470005	−0.235002	+	0.971995i	$0.575510\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	95089.7i	0.983134i	0.870840	+	0.491567i	$0.163576\pi$
$311$	95089.7i	0.983134i	−0.870840	+	0.491567i	$0.836424\pi$
$312$	0	0
$313$	−170572.	−1.74108	−0.870541	−	0.492097i	$-0.836231\pi$
$313$	−170572.	−1.74108	−0.870541	+	0.492097i	$0.836231\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 121998.i	− 1.21405i	−0.794684	−	0.607024i	$-0.792363\pi$
$317$	− 121998.i	− 1.21405i	0.794684	−	0.607024i	$-0.207637\pi$
$318$	0	0
$319$	135671.	1.33323
$320$	0	0
$321$	0	0
$322$	0	0
$323$	241563.i	2.31540i
$324$	0	0
$325$	35672.9	0.337731
$326$	0	0
$327$	0	0
$328$	0	0
$329$	6150.28i	0.0568203i
$330$	0	0
$331$	95214.2	0.869052	0.434526	−	0.900659i	$-0.356916\pi$
$331$	95214.2	0.869052	0.434526	+	0.900659i	$0.356916\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	16615.2i	0.148053i
$336$	0	0
$337$	8409.77	0.0740499	0.0370249	−	0.999314i	$-0.488212\pi$
$337$	8409.77	0.0740499	0.0370249	+	0.999314i	$0.488212\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 107194.i	− 0.921852i
$342$	0	0
$343$	−65677.6	−0.558250
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 6032.75i	− 0.0501022i	−0.999686	−	0.0250511i	$-0.992025\pi$
$347$	− 6032.75i	− 0.0501022i	0.999686	−	0.0250511i	$-0.00797484\pi$
$348$	0	0
$349$	139600.	1.14613	0.573065	−	0.819510i	$-0.305754\pi$
$349$	139600.	1.14613	0.573065	+	0.819510i	$0.305754\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	85395.3i	0.685306i	0.939462	+	0.342653i	$0.111326\pi$
$353$	85395.3i	0.685306i	−0.939462	+	0.342653i	$0.888674\pi$
$354$	0	0
$355$	−69778.6	−0.553689
$356$	0	0
$357$	0	0
$358$	0	0
$359$	90712.4i	0.703846i	0.936029	+	0.351923i	$0.114472\pi$
$359$	90712.4i	0.703846i	−0.936029	+	0.351923i	$0.885528\pi$
$360$	0	0
$361$	368579.	2.82824
$362$	0	0
$363$	0	0
$364$	0	0
$365$	52032.6i	0.390562i
$366$	0	0
$367$	25541.3	0.189632	0.0948160	−	0.995495i	$-0.469774\pi$
$367$	25541.3	0.189632	0.0948160	+	0.995495i	$0.469774\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 75297.9i	− 0.547060i
$372$	0	0
$373$	−234528.	−1.68569	−0.842844	−	0.538158i	$-0.819120\pi$
$373$	−234528.	−1.68569	−0.842844	+	0.538158i	$0.819120\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 97984.7i	− 0.689407i
$378$	0	0
$379$	107483.	0.748278	0.374139	−	0.927373i	$-0.377938\pi$
$379$	107483.	0.748278	0.374139	+	0.927373i	$0.377938\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	164427.i	1.12093i	0.828180	+	0.560463i	$0.189377\pi$
$383$	164427.i	1.12093i	−0.828180	+	0.560463i	$0.810623\pi$
$384$	0	0
$385$	−18275.2	−0.123294
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 182900.i	− 1.20869i	−0.796724	−	0.604343i	$-0.793435\pi$
$389$	− 182900.i	− 1.20869i	0.796724	−	0.604343i	$-0.206565\pi$
$390$	0	0
$391$	−203937.	−1.33396
$392$	0	0
$393$	0	0
$394$	0	0
$395$	75354.6i	0.482965i
$396$	0	0
$397$	−277717.	−1.76206	−0.881031	−	0.473059i	$-0.843150\pi$
$397$	−277717.	−1.76206	−0.881031	+	0.473059i	$0.843150\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 168062.i	− 1.04515i	−0.852593	−	0.522576i	$-0.824971\pi$
$401$	− 168062.i	− 1.04515i	0.852593	−	0.522576i	$-0.175029\pi$
$402$	0	0
$403$	−77418.0	−0.476685
$404$	0	0
$405$	0	0
$406$	0	0
$407$	58661.9i	0.354134i
$408$	0	0
$409$	−88381.1	−0.528339	−0.264170	−	0.964476i	$-0.585098\pi$
$409$	−88381.1	−0.528339	−0.264170	+	0.964476i	$0.585098\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	80551.4i	0.472251i
$414$	0	0
$415$	−92235.5	−0.535552
$416$	0	0
$417$	0	0
$418$	0	0
$419$	151429.i	0.862542i	0.902222	+	0.431271i	$0.141935\pi$
$419$	151429.i	0.862542i	−0.902222	+	0.431271i	$0.858065\pi$
$420$	0	0
$421$	−132220.	−0.745992	−0.372996	−	0.927833i	$-0.621669\pi$
$421$	−132220.	−0.745992	−0.372996	+	0.927833i	$0.621669\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	162160.i	0.897772i
$426$	0	0
$427$	16158.9	0.0886247
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 231482.i	− 1.24613i	−0.782170	−	0.623065i	$-0.785887\pi$
$431$	− 231482.i	− 1.24613i	0.782170	−	0.623065i	$-0.214113\pi$
$432$	0	0
$433$	−218090.	−1.16321	−0.581607	−	0.813470i	$-0.697576\pi$
$433$	−218090.	−1.16321	−0.581607	+	0.813470i	$0.697576\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	421192.i	2.20555i
$438$	0	0
$439$	−89557.0	−0.464698	−0.232349	−	0.972633i	$-0.574641\pi$
$439$	−89557.0	−0.464698	−0.232349	+	0.972633i	$0.574641\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	264286.i	1.34669i	0.739330	+	0.673343i	$0.235142\pi$
$443$	264286.i	1.34669i	−0.739330	+	0.673343i	$0.764858\pi$
$444$	0	0
$445$	107119.	0.540936
$446$	0	0
$447$	0	0
$448$	0	0
$449$	33967.2i	0.168487i	0.996445	+	0.0842435i	$0.0268474\pi$
$449$	33967.2i	0.168487i	−0.996445	+	0.0842435i	$0.973153\pi$
$450$	0	0
$451$	−10328.0	−0.0507765
$452$	0	0
$453$	0	0
$454$	0	0
$455$	13198.8i	0.0637546i
$456$	0	0
$457$	83273.7	0.398727	0.199363	−	0.979926i	$-0.436113\pi$
$457$	83273.7	0.398727	0.199363	+	0.979926i	$0.436113\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	57510.4i	0.270610i	0.990804	+	0.135305i	$0.0432015\pi$
$461$	57510.4i	0.270610i	−0.990804	+	0.135305i	$0.956799\pi$
$462$	0	0
$463$	−81793.9	−0.381557	−0.190778	−	0.981633i	$-0.561101\pi$
$463$	−81793.9	−0.381557	−0.190778	+	0.981633i	$0.561101\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 161762.i	− 0.741726i	−0.928688	−	0.370863i	$-0.879062\pi$
$467$	− 161762.i	− 0.741726i	0.928688	−	0.370863i	$-0.120938\pi$
$468$	0	0
$469$	19323.8	0.0878511
$470$	0	0
$471$	0	0
$472$	0	0
$473$	93390.6i	0.417427i
$474$	0	0
$475$	334910.	1.48436
$476$	0	0
$477$	0	0
$478$	0	0
$479$	128459.i	0.559878i	0.960018	+	0.279939i	$0.0903142\pi$
$479$	128459.i	0.559878i	−0.960018	+	0.279939i	$0.909686\pi$
$480$	0	0
$481$	42367.0	0.183121
$482$	0	0
$483$	0	0
$484$	0	0
$485$	66829.5i	0.284109i
$486$	0	0
$487$	255021.	1.07527	0.537636	−	0.843177i	$-0.319318\pi$
$487$	255021.	1.07527	0.537636	+	0.843177i	$0.319318\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	174548.i	0.724023i	0.932174	+	0.362012i	$0.117910\pi$
$491$	174548.i	0.724023i	−0.932174	+	0.362012i	$0.882090\pi$
$492$	0	0
$493$	445414.	1.83261
$494$	0	0
$495$	0	0
$496$	0	0
$497$	81153.7i	0.328546i
$498$	0	0
$499$	−209729.	−0.842282	−0.421141	−	0.906995i	$-0.638370\pi$
$499$	−209729.	−0.842282	−0.421141	+	0.906995i	$0.638370\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 288148.i	− 1.13888i	−0.822031	−	0.569442i	$-0.807159\pi$
$503$	− 288148.i	− 1.13888i	0.822031	−	0.569442i	$-0.192841\pi$
$504$	0	0
$505$	91900.8	0.360360
$506$	0	0
$507$	0	0
$508$	0	0
$509$	84680.9i	0.326851i	0.986556	+	0.163426i	$0.0522544\pi$
$509$	84680.9i	0.326851i	−0.986556	+	0.163426i	$0.947746\pi$
$510$	0	0
$511$	60514.9	0.231750
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 166746.i	− 0.628695i
$516$	0	0
$517$	−44852.8	−0.167806
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 80148.3i	− 0.295270i	−0.989042	−	0.147635i	$-0.952834\pi$
$521$	− 80148.3i	− 0.295270i	0.989042	−	0.147635i	$-0.0471660\pi$
$522$	0	0
$523$	−127783.	−0.467163	−0.233581	−	0.972337i	$-0.575045\pi$
$523$	−127783.	−0.467163	−0.233581	+	0.972337i	$0.575045\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 351923.i	− 1.26715i
$528$	0	0
$529$	−75747.1	−0.270679
$530$	0	0
$531$	0	0
$532$	0	0
$533$	7459.12i	0.0262563i
$534$	0	0
$535$	205620.	0.718385
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 228860.i	− 0.787756i
$540$	0	0
$541$	1167.97	0.00399060	0.00199530	−	0.999998i	$-0.499365\pi$
$541$	1167.97	0.00399060	0.00199530	+	0.999998i	$0.499365\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 148226.i	− 0.499035i
$546$	0	0
$547$	−277805.	−0.928465	−0.464232	−	0.885713i	$-0.653670\pi$
$547$	−277805.	−0.928465	−0.464232	+	0.885713i	$0.653670\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 919914.i	− 3.03001i
$552$	0	0
$553$	87638.7	0.286580
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 127126.i	− 0.409755i	−0.978788	−	0.204877i	$-0.934320\pi$
$557$	− 127126.i	− 0.409755i	0.978788	−	0.204877i	$-0.0656795\pi$
$558$	0	0
$559$	67448.9	0.215850
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 123792.i	− 0.390550i	−0.980749	−	0.195275i	$-0.937440\pi$
$563$	− 123792.i	− 0.390550i	0.980749	−	0.195275i	$-0.0625599\pi$
$564$	0	0
$565$	−8653.83	−0.0271089
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 61812.5i	− 0.190920i	−0.995433	−	0.0954601i	$-0.969568\pi$
$569$	− 61812.5i	− 0.190920i	0.995433	−	0.0954601i	$-0.0304322\pi$
$570$	0	0
$571$	204135.	0.626101	0.313050	−	0.949737i	$-0.398649\pi$
$571$	204135.	0.626101	0.313050	+	0.949737i	$0.398649\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	282745.i	0.855182i
$576$	0	0
$577$	555504.	1.66854	0.834269	−	0.551358i	$-0.185890\pi$
$577$	555504.	1.66854	0.834269	+	0.551358i	$0.185890\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	107272.i	0.317784i
$582$	0	0
$583$	549133.	1.61563
$584$	0	0
$585$	0	0
$586$	0	0
$587$	7016.77i	0.0203639i	0.999948	+	0.0101820i	$0.00324107\pi$
$587$	7016.77i	0.0203639i	−0.999948	+	0.0101820i	$0.996759\pi$
$588$	0	0
$589$	−726827.	−2.09508
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 77064.9i	− 0.219153i	−0.993978	−	0.109576i	$-0.965051\pi$
$593$	− 77064.9i	− 0.219153i	0.993978	−	0.109576i	$-0.0349494\pi$
$594$	0	0
$595$	−59998.4	−0.169475
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 546962.i	− 1.52442i	−0.647333	−	0.762208i	$-0.724116\pi$
$599$	− 546962.i	− 1.52442i	0.647333	−	0.762208i	$-0.275884\pi$
$600$	0	0
$601$	−539976.	−1.49495	−0.747473	−	0.664292i	$-0.768733\pi$
$601$	−539976.	−1.49495	−0.747473	+	0.664292i	$0.768733\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	46541.6i	0.127154i
$606$	0	0
$607$	−499326.	−1.35521	−0.677606	−	0.735425i	$-0.736982\pi$
$607$	−499326.	−1.35521	−0.677606	+	0.735425i	$0.736982\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	32393.8i	0.0867719i
$612$	0	0
$613$	80551.4	0.214364	0.107182	−	0.994239i	$-0.465817\pi$
$613$	80551.4	0.214364	0.107182	+	0.994239i	$0.465817\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	444845.i	1.16853i	0.811564	+	0.584263i	$0.198617\pi$
$617$	444845.i	1.16853i	−0.811564	+	0.584263i	$0.801383\pi$
$618$	0	0
$619$	−560941.	−1.46398	−0.731992	−	0.681314i	$-0.761409\pi$
$619$	−560941.	−1.46398	−0.731992	+	0.681314i	$0.761409\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 124581.i	− 0.320979i
$624$	0	0
$625$	130545.	0.334196
$626$	0	0
$627$	0	0
$628$	0	0
$629$	192590.i	0.486780i
$630$	0	0
$631$	−621772.	−1.56161	−0.780805	−	0.624775i	$-0.785191\pi$
$631$	−621772.	−1.56161	−0.780805	+	0.624775i	$0.785191\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	197125.i	0.488871i
$636$	0	0
$637$	−165288.	−0.407345
$638$	0	0
$639$	0	0
$640$	0	0
$641$	322644.i	0.785248i	0.919699	+	0.392624i	$0.128433\pi$
$641$	322644.i	0.785248i	−0.919699	+	0.392624i	$0.871567\pi$
$642$	0	0
$643$	489813.	1.18470	0.592350	−	0.805681i	$-0.298200\pi$
$643$	489813.	1.18470	0.592350	+	0.805681i	$0.298200\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 26885.7i	− 0.0642263i	−0.999484	−	0.0321132i	$-0.989776\pi$
$647$	− 26885.7i	− 0.0642263i	0.999484	−	0.0321132i	$-0.0102237\pi$
$648$	0	0
$649$	−587446.	−1.39469
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 714229.i	− 1.67498i	−0.546449	−	0.837492i	$-0.684021\pi$
$653$	− 714229.i	− 1.67498i	0.546449	−	0.837492i	$-0.315979\pi$
$654$	0	0
$655$	−190422.	−0.443849
$656$	0	0
$657$	0	0
$658$	0	0
$659$	234697.i	0.540428i	0.962800	+	0.270214i	$0.0870944\pi$
$659$	234697.i	0.540428i	−0.962800	+	0.270214i	$0.912906\pi$
$660$	0	0
$661$	626562.	1.43404	0.717020	−	0.697053i	$-0.245506\pi$
$661$	626562.	1.43404	0.717020	+	0.697053i	$0.245506\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	123915.i	0.280208i
$666$	0	0
$667$	776630.	1.74567
$668$	0	0
$669$	0	0
$670$	0	0
$671$	117843.i	0.261734i
$672$	0	0
$673$	641998.	1.41744	0.708718	−	0.705491i	$-0.249274\pi$
$673$	641998.	1.41744	0.708718	+	0.705491i	$0.249274\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 656946.i	− 1.43335i	−0.697407	−	0.716675i	$-0.745663\pi$
$677$	− 656946.i	− 1.43335i	0.697407	−	0.716675i	$-0.254337\pi$
$678$	0	0
$679$	77723.9	0.168583
$680$	0	0
$681$	0	0
$682$	0	0
$683$	76710.6i	0.164442i	0.996614	+	0.0822212i	$0.0262014\pi$
$683$	76710.6i	0.164442i	−0.996614	+	0.0822212i	$0.973799\pi$
$684$	0	0
$685$	−10938.4	−0.0233117
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 396597.i	− 0.835432i
$690$	0	0
$691$	570869.	1.19559	0.597793	−	0.801651i	$-0.296045\pi$
$691$	570869.	1.19559	0.597793	+	0.801651i	$0.296045\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 114936.i	− 0.237951i
$696$	0	0
$697$	−33907.3	−0.0697956
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 4271.71i	− 0.00869292i	−0.999991	−	0.00434646i	$-0.998616\pi$
$701$	− 4271.71i	− 0.00869292i	0.999991	−	0.00434646i	$-0.00138353\pi$
$702$	0	0
$703$	397757.	0.804835
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 106882.i	− 0.213829i
$708$	0	0
$709$	148443.	0.295303	0.147652	−	0.989039i	$-0.452829\pi$
$709$	148443.	0.295303	0.147652	+	0.989039i	$0.452829\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 613618.i	− 1.20703i
$714$	0	0
$715$	−96256.2	−0.188285
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 549881.i	− 1.06368i	−0.846845	−	0.531840i	$-0.821501\pi$
$719$	− 549881.i	− 1.06368i	0.846845	−	0.531840i	$-0.178499\pi$
$720$	0	0
$721$	−193928.	−0.373052
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 617534.i	− 1.17486i
$726$	0	0
$727$	−644007.	−1.21849	−0.609244	−	0.792982i	$-0.708527\pi$
$727$	−644007.	−1.21849	−0.609244	+	0.792982i	$0.708527\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	306606.i	0.573780i
$732$	0	0
$733$	832831.	1.55006	0.775031	−	0.631924i	$-0.217734\pi$
$733$	832831.	1.55006	0.775031	+	0.631924i	$0.217734\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	140925.i	0.259449i
$738$	0	0
$739$	−472487.	−0.865170	−0.432585	−	0.901593i	$-0.642398\pi$
$739$	−472487.	−0.865170	−0.432585	+	0.901593i	$0.642398\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 978422.i	− 1.77235i	−0.463354	−	0.886173i	$-0.653354\pi$
$743$	− 978422.i	− 1.77235i	0.463354	−	0.886173i	$-0.346646\pi$
$744$	0	0
$745$	−499497.	−0.899954
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 239139.i	− 0.426273i
$750$	0	0
$751$	166281.	0.294824	0.147412	−	0.989075i	$-0.452906\pi$
$751$	166281.	0.294824	0.147412	+	0.989075i	$0.452906\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	306733.i	0.538105i
$756$	0	0
$757$	−238453.	−0.416114	−0.208057	−	0.978117i	$-0.566714\pi$
$757$	−238453.	−0.416114	−0.208057	+	0.978117i	$0.566714\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 598335.i	− 1.03318i	−0.856233	−	0.516589i	$-0.827201\pi$
$761$	− 598335.i	− 1.03318i	0.856233	−	0.516589i	$-0.172799\pi$
$762$	0	0
$763$	−172389.	−0.296116
$764$	0	0
$765$	0	0
$766$	0	0
$767$	424267.i	0.721189i
$768$	0	0
$769$	667323.	1.12845	0.564227	−	0.825620i	$-0.309174\pi$
$769$	667323.	1.12845	0.564227	+	0.825620i	$0.309174\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 848153.i	− 1.41943i	−0.704487	−	0.709717i	$-0.748823\pi$
$773$	− 848153.i	− 1.41943i	0.704487	−	0.709717i	$-0.251177\pi$
$774$	0	0
$775$	−487916.	−0.812347
$776$	0	0
$777$	0	0
$778$	0	0
$779$	70028.8i	0.115399i
$780$	0	0
$781$	−591838.	−0.970289
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 404300.i	− 0.656091i
$786$	0	0
$787$	967733.	1.56245	0.781225	−	0.624250i	$-0.214595\pi$
$787$	967733.	1.56245	0.781225	+	0.624250i	$0.214595\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	10064.6i	0.0160858i
$792$	0	0
$793$	85109.3	0.135341
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 674146.i	− 1.06130i	−0.847592	−	0.530649i	$-0.821948\pi$
$797$	− 674146.i	− 1.06130i	0.847592	−	0.530649i	$-0.178052\pi$
$798$	0	0
$799$	−147254.	−0.230661
$800$	0	0
$801$	0	0
$802$	0	0
$803$	441323.i	0.684425i
$804$	0	0
$805$	−104614.	−0.161435
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 570317.i	− 0.871403i	−0.900091	−	0.435702i	$-0.856500\pi$
$809$	− 570317.i	− 0.871403i	0.900091	−	0.435702i	$-0.143500\pi$
$810$	0	0
$811$	111731.	0.169876	0.0849379	−	0.996386i	$-0.472931\pi$
$811$	111731.	0.169876	0.0849379	+	0.996386i	$0.472931\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 169444.i	− 0.255100i
$816$	0	0
$817$	633234.	0.948680
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1.04162e6i	1.54534i	0.634811	+	0.772668i	$0.281078\pi$
$821$	1.04162e6i	1.54534i	−0.634811	+	0.772668i	$0.718922\pi$
$822$	0	0
$823$	108973.	0.160886	0.0804432	−	0.996759i	$-0.474366\pi$
$823$	108973.	0.160886	0.0804432	+	0.996759i	$0.474366\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	550908.i	0.805505i	0.915309	+	0.402753i	$0.131947\pi$
$827$	550908.i	0.805505i	−0.915309	+	0.402753i	$0.868053\pi$
$828$	0	0
$829$	−57256.0	−0.0833128	−0.0416564	−	0.999132i	$-0.513263\pi$
$829$	−57256.0	−0.0833128	−0.0416564	+	0.999132i	$0.513263\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 751358.i	− 1.08282i
$834$	0	0
$835$	139183.	0.199624
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 951705.i	− 1.35201i	−0.736899	−	0.676003i	$-0.763711\pi$
$839$	− 951705.i	− 1.35201i	0.736899	−	0.676003i	$-0.236289\pi$
$840$	0	0
$841$	−988934.	−1.39822
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 281264.i	− 0.393914i
$846$	0	0
$847$	54128.7	0.0754502
$848$	0	0
$849$	0	0
$850$	0	0
$851$	335803.i	0.463687i
$852$	0	0
$853$	−1.02287e6	−1.40579	−0.702896	−	0.711293i	$-0.748110\pi$
$853$	−1.02287e6	−1.40579	−0.702896	+	0.711293i	$0.748110\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 661433.i	− 0.900584i	−0.892881	−	0.450292i	$-0.851320\pi$
$857$	− 661433.i	− 0.900584i	0.892881	−	0.450292i	$-0.148680\pi$
$858$	0	0
$859$	−102749.	−0.139249	−0.0696246	−	0.997573i	$-0.522180\pi$
$859$	−102749.	−0.139249	−0.0696246	+	0.997573i	$0.522180\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 168806.i	− 0.226655i	−0.993558	−	0.113328i	$-0.963849\pi$
$863$	− 168806.i	− 0.226655i	0.993558	−	0.113328i	$-0.0361509\pi$
$864$	0	0
$865$	−481087.	−0.642971
$866$	0	0
$867$	0	0
$868$	0	0
$869$	639132.i	0.846352i
$870$	0	0
$871$	101779.	0.134160
$872$	0	0
$873$	0	0
$874$	0	0
$875$	192830.i	0.251860i
$876$	0	0
$877$	−190156.	−0.247235	−0.123618	−	0.992330i	$-0.539450\pi$
$877$	−190156.	−0.247235	−0.123618	+	0.992330i	$0.539450\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 171490.i	− 0.220946i	−0.993879	−	0.110473i	$-0.964763\pi$
$881$	− 171490.i	− 0.220946i	0.993879	−	0.110473i	$-0.0352366\pi$
$882$	0	0
$883$	−869284.	−1.11491	−0.557456	−	0.830207i	$-0.688222\pi$
$883$	−869284.	−1.11491	−0.557456	+	0.830207i	$0.688222\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 144353.i	− 0.183476i	−0.995783	−	0.0917378i	$-0.970758\pi$
$887$	− 144353.i	− 0.183476i	0.995783	−	0.0917378i	$-0.0292422\pi$
$888$	0	0
$889$	229260.	0.290084
$890$	0	0
$891$	0	0
$892$	0	0
$893$	304124.i	0.381371i
$894$	0	0
$895$	−295961.	−0.369477
$896$	0	0
$897$	0	0
$898$	0	0
$899$	1.34018e6i	1.65823i
$900$	0	0
$901$	1.80283e6	2.22078
$902$	0	0
$903$	0	0
$904$	0	0
$905$	126222.i	0.154113i
$906$	0	0
$907$	−107697.	−0.130915	−0.0654574	−	0.997855i	$-0.520851\pi$
$907$	−107697.	−0.130915	−0.0654574	+	0.997855i	$0.520851\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 712977.i	− 0.859091i	−0.903045	−	0.429545i	$-0.858674\pi$
$911$	− 712977.i	− 0.859091i	0.903045	−	0.429545i	$-0.141326\pi$
$912$	0	0
$913$	−782310.	−0.938507
$914$	0	0
$915$	0	0
$916$	0	0
$917$	221465.i	0.263370i
$918$	0	0
$919$	−239512.	−0.283594	−0.141797	−	0.989896i	$-0.545288\pi$
$919$	−239512.	−0.283594	−0.141797	+	0.989896i	$0.545288\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	427440.i	0.501732i
$924$	0	0
$925$	267012.	0.312067
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 1.28407e6i	− 1.48785i	−0.668266	−	0.743923i	$-0.732963\pi$
$929$	− 1.28407e6i	− 1.48785i	0.668266	−	0.743923i	$-0.267037\pi$
$930$	0	0
$931$	−1.55178e6	−1.79032
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 437557.i	− 0.500508i
$936$	0	0
$937$	−471000.	−0.536466	−0.268233	−	0.963354i	$-0.586440\pi$
$937$	−471000.	−0.536466	−0.268233	+	0.963354i	$0.586440\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	721460.i	0.814766i	0.913257	+	0.407383i	$0.133559\pi$
$941$	721460.i	0.814766i	−0.913257	+	0.407383i	$0.866441\pi$
$942$	0	0
$943$	−59121.2	−0.0664845
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 605552.i	− 0.675229i	−0.941284	−	0.337615i	$-0.890380\pi$
$947$	− 605552.i	− 0.675229i	0.941284	−	0.337615i	$-0.109620\pi$
$948$	0	0
$949$	318734.	0.353913
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 527043.i	− 0.580310i	−0.956980	−	0.290155i	$-0.906293\pi$
$953$	− 527043.i	− 0.580310i	0.956980	−	0.290155i	$-0.0937069\pi$
$954$	0	0
$955$	430240.	0.471741
$956$	0	0
$957$	0	0
$958$	0	0
$959$	12721.6i	0.0138326i
$960$	0	0
$961$	135363.	0.146573
$962$	0	0
$963$	0	0
$964$	0	0
$965$	64363.8i	0.0691174i
$966$	0	0
$967$	1.24031e6	1.32641	0.663203	−	0.748439i	$-0.269196\pi$
$967$	1.24031e6	1.32641	0.663203	+	0.748439i	$0.269196\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 1.27298e6i	− 1.35015i	−0.737748	−	0.675076i	$-0.764111\pi$
$971$	− 1.27298e6i	− 1.35015i	0.737748	−	0.675076i	$-0.235889\pi$
$972$	0	0
$973$	−133673.	−0.141194
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1.38469e6i	1.45065i	0.688406	+	0.725325i	$0.258311\pi$
$977$	1.38469e6i	1.45065i	−0.688406	+	0.725325i	$0.741689\pi$
$978$	0	0
$979$	908546.	0.947942
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 169030.i	− 0.174927i	−0.996168	−	0.0874633i	$-0.972124\pi$
$983$	− 169030.i	− 0.174927i	0.996168	−	0.0874633i	$-0.0278761\pi$
$984$	0	0
$985$	−521016.	−0.537006
$986$	0	0
$987$	0	0
$988$	0	0
$989$	534602.i	0.546560i
$990$	0	0
$991$	1.57481e6	1.60354	0.801772	−	0.597630i	$-0.203891\pi$
$991$	1.57481e6	1.60354	0.801772	+	0.597630i	$0.203891\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 384055.i	− 0.387924i
$996$	0	0
$997$	570747.	0.574187	0.287094	−	0.957903i	$-0.407311\pi$
$997$	570747.	0.574187	0.287094	+	0.957903i	$0.407311\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1296.5.e.g.161.6		8
3.2	odd	2	inner	1296.5.e.g.161.3		8
4.3	odd	2		324.5.c.a.161.6		8
9.2	odd	6		144.5.q.c.113.1		8
9.4	even	3		144.5.q.c.65.1		8
9.5	odd	6		432.5.q.c.305.2		8
9.7	even	3		432.5.q.c.17.2		8
12.11	even	2		324.5.c.a.161.3		8
36.7	odd	6		108.5.g.a.17.2		8
36.11	even	6		36.5.g.a.5.4	✓	8
36.23	even	6		108.5.g.a.89.2		8
36.31	odd	6		36.5.g.a.29.4	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
36.5.g.a.5.4	✓	8	36.11	even	6
36.5.g.a.29.4	yes	8	36.31	odd	6
108.5.g.a.17.2		8	36.7	odd	6
108.5.g.a.89.2		8	36.23	even	6
144.5.q.c.65.1		8	9.4	even	3
144.5.q.c.113.1		8	9.2	odd	6
324.5.c.a.161.3		8	12.11	even	2
324.5.c.a.161.6		8	4.3	odd	2
432.5.q.c.17.2		8	9.7	even	3
432.5.q.c.305.2		8	9.5	odd	6
1296.5.e.g.161.3		8	3.2	odd	2	inner
1296.5.e.g.161.6		8	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1296 = 2^{4} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	1296.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+12.2819i q^{5} +14.2840 q^{7} +O(q^{10})\)	\(q+12.2819i q^{5} +14.2840 q^{7} +104.171i q^{11} +75.2346 q^{13} +341.998i q^{17} +706.329 q^{19} +596.312i q^{23} +474.155 q^{25} -1302.39i q^{29} -1029.02 q^{31} +175.435i q^{35} +563.132 q^{37} +99.1448i q^{41} +896.514 q^{43} +430.570i q^{47} -2196.97 q^{49} -5271.47i q^{53} -1279.41 q^{55} +5639.26i q^{59} +1131.25 q^{61} +924.023i q^{65} +1352.82 q^{67} +5681.42i q^{71} +4236.54 q^{73} +1487.98i q^{77} +6135.42 q^{79} +7509.88i q^{83} -4200.38 q^{85} -8721.70i q^{89} +1074.65 q^{91} +8675.05i q^{95} +5441.31 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 26 q^{7}+O(q^{10}) \) 8 * q + 26 * q^7	\( 8 q + 26 q^{7} + 10 q^{13} - 562 q^{19} - 706 q^{25} + 374 q^{31} + 16 q^{37} - 136 q^{43} + 654 q^{49} + 1818 q^{55} + 3874 q^{61} + 308 q^{67} - 7802 q^{73} - 4390 q^{79} + 6084 q^{85} - 15830 q^{91} - 14564 q^{97}+O(q^{100}) \) 8 * q + 26 * q^7 + 10 * q^13 - 562 * q^19 - 706 * q^25 + 374 * q^31 + 16 * q^37 - 136 * q^43 + 654 * q^49 + 1818 * q^55 + 3874 * q^61 + 308 * q^67 - 7802 * q^73 - 4390 * q^79 + 6084 * q^85 - 15830 * q^91 - 14564 * q^97

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